1,193 research outputs found

    Relating Graph Thickness to Planar Layers and Bend Complexity

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    The thickness of a graph G=(V,E)G=(V,E) with nn vertices is the minimum number of planar subgraphs of GG whose union is GG. A polyline drawing of GG in R2\mathbb{R}^2 is a drawing Γ\Gamma of GG, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Γ\Gamma is the maximum number of bends per edge in Γ\Gamma, and the layer complexity of Γ\Gamma is the minimum integer rr such that the set of polygonal chains in Γ\Gamma can be partitioned into rr disjoint sets, where each set corresponds to a planar polyline drawing. Let GG be a graph of thickness tt. By F\'{a}ry's theorem, if t=1t=1, then GG can be drawn on a single layer with bend complexity 00. A few extensions to higher thickness are known, e.g., if t=2t=2 (resp., t>2t>2), then GG can be drawn on tt layers with bend complexity 2 (resp., 3n+O(1)3n+O(1)). However, allowing a higher number of layers may reduce the bend complexity, e.g., complete graphs require Θ(n)\Theta(n) layers to be drawn using 0 bends per edge. In this paper we present an elegant extension of F\'{a}ry's theorem to draw graphs of thickness t>2t>2. We first prove that thickness-tt graphs can be drawn on tt layers with 2.25n+O(1)2.25n+O(1) bends per edge. We then develop another technique to draw thickness-tt graphs on tt layers with bend complexity, i.e., O(2t⋅n1−(1/β))O(\sqrt{2}^{t} \cdot n^{1-(1/\beta)}), where β=2⌈(t−2)/2⌉\beta = 2^{\lceil (t-2)/2 \rceil }. Previously, the bend complexity was not known to be sublinear for t>2t>2. Finally, we show that graphs with linear arboricity kk can be drawn on kk layers with bend complexity 3(k−1)n(4k−2)\frac{3(k-1)n}{(4k-2)}.Comment: A preliminary version appeared at the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016

    Snapping Graph Drawings to the Grid Optimally

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    In geographic information systems and in the production of digital maps for small devices with restricted computational resources one often wants to round coordinates to a rougher grid. This removes unnecessary detail and reduces space consumption as well as computation time. This process is called snapping to the grid and has been investigated thoroughly from a computational-geometry perspective. In this paper we investigate the same problem for given drawings of planar graphs under the restriction that their combinatorial embedding must be kept and edges are drawn straight-line. We show that the problem is NP-hard for several objectives and provide an integer linear programming formulation. Given a plane graph G and a positive integer w, our ILP can also be used to draw G straight-line on a grid of width w and minimum height (if possible).Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Straightening out planar poly-line drawings

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    We show that any yy-monotone poly-line drawing can be straightened out while maintaining yy-coordinates and height. The width may increase much, but we also show that on some graphs exponential width is required if we do not want to increase the height. Likewise yy-monotonicity is required: there are poly-line drawings (not yy-monotone) that cannot be straightened out while maintaining the height. We give some applications of our result.Comment: The main result turns out to be known (Pach & Toth, J. Graph Theory 2004, http://onlinelibrary.wiley.com/doi/10.1002/jgt.10168/pdf

    On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

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    We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

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    A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is xx- and yy-monotone. Angle-monotone graphs are 2\sqrt 2-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-θ6\theta_6-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex ss to any vertex tt whose length is within 1+21 + \sqrt 2 times the Euclidean distance from ss to tt. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Strictly convex drawings of planar graphs

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    Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma

    Drawing Planar Graphs with Few Geometric Primitives

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    We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let nn denote the number of vertices of a graph. We show that trees can be drawn with 3n/43n/4 straight-line segments on a polynomial grid, and with n/2n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n−17)/3(8n-17)/3 segments on an O(n)×O(n2)O(n)\times O(n^2) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/23n/2 edges on an O(n)×O(n2)O(n)\times O(n^2) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n−11)/3(5n - 11)/3 arcs. This is significantly smaller than the lower bound of 2n2n for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017
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